How to find the slant asymptote of #f(x) = (2x^2 +3) / (x-1)# ??

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Answer 1 · 2018-08-01T22:01:35

Essentially by division...

Given:

#f(x) = (2x^2+3)/(x-1)#

we can divide to get a polynomial quotient and rational remainder as follows:

#(2x^2+3)/(x-1) = (2x^2-2x+2x-2+5)/(x-1)#

#color(white)((2x^2+3)/(x-1)) = (2x(x-1)+2(x-1)+5)/(x-1)#

#color(white)((2x^2+3)/(x-1)) = 2x+2+5/(x-1)#

Then note that:

#lim_(x->oo) 5/(x-1) = 0#

#lim_(x->-oo) 5/(x-1) = 0#

So #2x+2# is the slant asymptote of #f(x) = (2x^2+3)/(x-1)#

graph{(y-(2x^2+3)/(x-1))(y-2x-2) = 0 [-42.17, 37.83, -16.48, 23.52]}